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On the statistical mechanics approach in the random matrix theory: Integrated density of states. (English) Zbl 1081.82569
Summary: We consider the ensemble of random symmetric \(n\times n\) matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit as \(n\to \infty\) and that this limiting distribution is the solution of a certain self-consistent equation.

MSC:
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
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